© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:

L10a100 L10a102

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a101 Visit L10a101's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X10,1,11,2 X12,3,13,4 X14,19,15,20 X18,7,19,8 X6,15,7,16 X16,5,17,6 X4,17,5,18 X20,13,9,14 X2,9,3,10 X8,11,1,12

Gauss Code:

{{1, -9, 2, -7, 6, -5, 4, -10}, {9, -1, 10, -2, 8, -3, 5, -6, 7, -4, 3, -8}}

Jones Polynomial:

- q-23/2 + 2q-21/2 - 4q-19/2 + 6q-17/2 - 8q-15/2 + 9q-13/2 - 9q-11/2 + 7q-9/2 - 6q-7/2 + 3q-5/2 - q-3/2

A2 (sl(3)) Invariant:

q-36 - q-32 + 2q-30 + 3q-24 + q-20 + 2q-14 - q-12 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial:

- a3z3 - 3a5z - 3a5z3 - a7z-1 - 3a7z - 3a7z3 + a9z-1 + a9z - a9z3 + a11z

Kauffman Polynomial:

- a3z3 - 3a4z4 - 3a5z + 6a5z3 - 6a5z5 - 3a6z2 + 9a6z4 - 7a6z6 - a7z-1 + 5a7z - 4a7z3 + 9a7z5 - 6a7z7 + a8 - 8a8z2 + 15a8z4 - a8z6 - 3a8z8 - a9z-1 + 7a9z - 18a9z3 + 21a9z5 - 5a9z7 - a9z9 - a10z2 - 9a10z4 + 15a10z6 - 5a10z8 + 3a11z - 15a11z3 + 11a11z5 - a11z9 + 4a12z2 - 12a12z4 + 9a12z6 - 2a12z8 + 4a13z - 8a13z3 + 5a13z5 - a13z7

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                     1 j = -4                   3 1 j = -6                 3     j = -8               4 3     j = -10             5 3       j = -12           4 4         j = -14         4 5           j = -16       2 4             j = -18     2 4               j = -20   1 3                 j = -22   1                   j = -24 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 101]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 101]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[14, 19, 15, 20], X[18, 7, 19, 8], > X[6, 15, 7, 16], X[16, 5, 17, 6], X[4, 17, 5, 18], X[20, 13, 9, 14], > X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -7, 6, -5, 4, -10}, > {9, -1, 10, -2, 8, -3, 5, -6, 7, -4, 3, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 4 6 8 9 9 7 6 -q + ----- - ----- + ----- - ----- + ----- - ----- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q q 3 -(3/2) > ---- - q 5/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 -32 2 3 -20 2 -12 2 -8 2 -4 q - q + --- + --- + q + --- - q + --- + q - -- + q 30 24 14 10 6 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 101]][a, z]
Out[8]=	7 9 a a 5 7 9 11 3 3 5 3 7 3 9 3 -(--) + -- - 3 a z - 3 a z + a z + a z - a z - 3 a z - 3 a z - a z z z
In[9]:=	Kauffman[Link[10, Alternating, 101]][a, z]
Out[9]=	7 9 8 a a 5 7 9 11 13 6 2 a - -- - -- - 3 a z + 5 a z + 7 a z + 3 a z + 4 a z - 3 a z - z z 8 2 10 2 12 2 3 3 5 3 7 3 9 3 > 8 a z - a z + 4 a z - a z + 6 a z - 4 a z - 18 a z - 11 3 13 3 4 4 6 4 8 4 10 4 > 15 a z - 8 a z - 3 a z + 9 a z + 15 a z - 9 a z - 12 4 5 5 7 5 9 5 11 5 13 5 6 6 > 12 a z - 6 a z + 9 a z + 21 a z + 11 a z + 5 a z - 7 a z - 8 6 10 6 12 6 7 7 9 7 13 7 8 8 > a z + 15 a z + 9 a z - 6 a z - 5 a z - a z - 3 a z - 10 8 12 8 9 9 11 9 > 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 3 2 4 2 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 24 10 22 9 20 9 20 8 18 8 18 7 16 7 q t q t q t q t q t q t q t 4 4 5 4 4 5 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 6 14 6 14 5 12 5 12 4 10 4 10 3 8 3 q t q t q t q t q t q t q t q t 3 3 3 > ----- + ----- + ---- 8 2 6 2 4 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a101

L10a100 L10a102